We recover Theorem REF , with a power saving error term of
\(O(X^{\frac{2}{7} + \epsilon })\) , by taking \(\alpha = 3\)  and \(\beta =4\)  and carrying out an appropriate calculation at the 2- and
3-adic places. (This was also noted in [1]}.) When
\(\frac{\beta }{\alpha } > \frac{7}{5}\) , the error term of
\(O(X^{\frac{5}{6\alpha }})\)  in case (1) dominates the other error term
and can be refined into a secondary term extrapolating that proved in
[2]}, [3]} for \(\alpha = 1\)  and \(\beta = 2\) . More precisely, we have the following result.
